Product inequalities for multivariate Gaussian, gamma, and positively upper orthant dependent distributions

The Gaussian product inequality is an important conjecture concerning the moments of Gaussian random vectors. While all attempts to prove the Gaussian product inequal-ity in full generality have been unsuccessful to date, numerous partial results have been derived in recent decades and we provide here further results on the problem. Most importantly, we establish a strong version of the Gaussian product inequality for multivariate gamma distributions in the case of nonnegative correlations, thereby extending a result recently derived by Genest and Ouimet (2021). Further, we show that the Gaussian product inequality holds with nonnegative exponents for all random vectors with positive components whenever the underlying vector is positively upper orthant dependent. Finally, we show that the Gaussian product inequality with negative exponents follows directly from the Gaussian correlation inequality.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

The Gaussian product inequality is an important conjecture about the moments of Gaussian random vectors. Although a full proof of the inequality has not been achieved, partial results have been obtained in recent decades. In this study, the authors provide further results on the problem, including a strong version of the inequality for multivariate gamma distributions with nonnegative correlations. The authors also show that the inequality holds for random vectors with positive components and nonnegative exponents when the underlying vector is positively upper orthant dependent.